On the extremality of Hofer's metric on the group of Hamiltonian diffeomorphisms

Abstract

Let M be a closed symplectic manifold, and let |·| be a norm on the space of all smooth functions on M, which are zero-mean normalized with respect to the canonical volume form. We show that if |·| ≤ C|·| ∞ and |·| is invariant under the action of Hamiltonian diffeomorphisms, then it is also invariant under all volume-preserving diffeomorphisms. We also prove that if |·| is, additionally, not equivalent to |·| ∞ , then the induced pseudodistance function on the group Ham (M,ω) of Hamiltonian diffeomorphisms of M vanishes identically. These results provide partial answers to questions raised by Eliashberg and Polterovich in 1993. Both results rely on an extension of |·| to the space of essentially bounded measurable functions, which is invariant under all measure-preserving bijections.